[HN Gopher] PDEs You Should Know
___________________________________________________________________
PDEs You Should Know
Author : lucaspauker
Score : 67 points
Date : 2022-02-08 19:32 UTC (3 hours ago)
(HTM) web link (www.lucaspauker.com)
(TXT) w3m dump (www.lucaspauker.com)
| dls2016 wrote:
| The Unfinished PDE Coffee Table Book
| https://people.maths.ox.ac.uk/trefethen/pdectb.html
| Koshkin wrote:
| This is much, much more useful (and beautiful, too).
| marginalia_nu wrote:
| PDEs are really useful if you are in the rare domains where they
| are useful. But most PDEs don't have even have closed form
| solutions for non-trivial boundary conditions. So unless you are
| a physicist or something adjacent, no, no you really don't need
| to know these.
|
| Hard to say what's the intended audience for the page though.
| Could be a message aimed at physics undergrads or something. If
| so, then indeed, you should know these.
| kleene_op wrote:
| Being familiar with PDEs is useful even without knowing how to
| solve them.
|
| Just the formulas alone teach you how physical quantities
| interact with one another and give you great insights on how
| the universe operates on a fundamental level.
|
| Most people may not be using them at their everyday job, or at
| all for that matter, but knowing the core ones is just as
| enlightening if not more than having read major works in
| Philosophy.
| kurthr wrote:
| As an engineer, I've used lots of FEA to solve problems, but if
| you can't put part of your solution space where it can be
| approximated by a closed form solution (and there are many more
| than those shown, which are amenable to solution to appropriate
| methods), you're going to have a hard time building a trusted
| model. There's a reason we still have wind tunnels.
|
| The most interesting parts of good FEA (where you've shown your
| model and reality match on measurable components), is that you
| can see hidden and unmeasurable variables, which may be design
| limiting.
| medo-bear wrote:
| Most useful equations don't have a closed form solution. This
| is why things like machine learning exist. More specifically
| PDEs are a hot topic in DNNs at the moment. See physics
| informed neural networks:
| https://en.m.wikipedia.org/wiki/Physics-informed_neural_netw...
| marginalia_nu wrote:
| The point is that equations without closed form solutions are
| pretty useless if you aren't into pretty hairy maths, Green's
| functions and the like.
|
| An equation everyone should know is Hooke's law. That's
| useful at a high school level.
| blablabla123 wrote:
| I think the most generic approach is to prove existence and
| uniqueness and then go with numerical methods.
| aqme28 wrote:
| I think we need a better mutual definition of useful. IMO
| an equation is "useful" if someone out there is doing
| practical things with it. You are taking a sort of
| definition where anyone can do math with it.
| aqme28 wrote:
| The math isn't _that_ hairy. I took numerical methods and
| computational physics classes in undergrad.
| [deleted]
| nautilius wrote:
| You really should read up on numerical math, we really
| don't need closed form solutions, and this makes PDE far
| from useless.
| medo-bear wrote:
| i disagree. equations are useful only if you have use for
| them. this is by definition of the term. on the other hand,
| high schoolers rarely feel like Hooke's law is useful!
| pvarangot wrote:
| Isn't Hooke's law a solution to the harmonic motion PDE in
| that page?
| tagoregrtst wrote:
| No?
|
| Hooks law is an approximation to a material (and spring)
| property that sets the PDE up. Sin(x) is the solution.
|
| But I could be wrong :S
| capn_duck wrote:
| A _Sin(w_ x) is a solution for certain initial
| conditions. In general there's no single "solution" to a
| PDE.
| aqme28 wrote:
| Why does a PDE need an analytical solution?
|
| Are you arguing that e.g. Navier-Stokes isn't useful?
|
| edit: Just noticed that Navier-Stokes isn't even on here. This
| is frankly a weird list.
| marginalia_nu wrote:
| Well, to be useful, wouldn't you need to be able to use it?
| Most people outside of physicists and physics-adjacent fields
| are very far away from the mathematical tools to deal with
| these equations.
| rprenger wrote:
| I think what the other commenters are getting at is that
| PDEs can be used without having a closed form solution (and
| mostly are used that way as closed form solutions usually
| only come up in special artificial cases). You start your
| system in a real known state and then propagate it forward
| in time using (for example) the finite difference method on
| the equations to figure out the state at a later time. http
| s://en.wikipedia.org/wiki/Numerical_methods_for_partial_...
| phkahler wrote:
| >> Well, to be useful, wouldn't you need to be able to use
| it?
|
| That's what we have computers for, numerical solutions to
| PDEs ;-)
| tagoregrtst wrote:
| Do you have to solve it to use it?
|
| NS is a case in point. No general solution, but thousands
| of special cases that are solved and many more that can be
| understood using numerical methods
| aqme28 wrote:
| Numerical methods exist. There's a whole field to simulate
| PDEs that you can't solve exactly.
| sampo wrote:
| > There's a whole field to simulate PDEs that you can't
| solve exactly.
|
| There are whole fields to simulate one particular PDE:
| Computational fluid mechanics for the Navier-Stokes
| equation. Computational electromagnetics for Maxwell's
| equations. Computational chemistry for the Schrodinger
| equation. Mathematical finance ...probably does also
| other things than just simulates the Black-Scholes
| equation.
| medo-bear wrote:
| cs is a dominant topic on hn. cs is definitely physics and
| math adjacent
| JadeNB wrote:
| > PDEs are really useful if you are in the rare domains where
| they are useful.
|
| Aside from 'rare', this seems at best vacuously true.
|
| > But most PDEs don't have even have closed form solutions for
| non-trivial boundary conditions. So unless you are a physicist
| or something adjacent, no, no you really don't need to know
| these.
|
| As others have said, while your first sentence is surely true,
| the latter doesn't follow from it (and I would argue isn't true
| --but it depends on how you define adjacency). There are lots
| of things one can usefully do with an equation besides finding
| a closed-form solution. (For an ODE example, the classical
| predator-prey model does not have a nice closed-form solution,
| but is still plenty useful.)
| travisporter wrote:
| Why do i need to enable javascript to see the equations tex-style
| Kwpolska wrote:
| This website is using MathJax [0] to render math. MathJax and
| its faster and leaner competitor KaTeX are the only ways to
| display beautiful, human-friendly math on the Web. They can be
| run server-side, but many sites do it client-side. The
| alternative, MathML [2] is a pain for humans to write [3] --
| it's a late-90s XML format -- and is only supported by Firefox
| and Safari [4].
|
| [0] https://www.mathjax.org/
|
| [1] https://katex.org/
|
| [2] https://en.wikipedia.org/wiki/MathML
|
| [3] https://fred-wang.github.io/MathFonts/mozilla_mathml_test/
|
| [4] https://developer.mozilla.org/en-US/docs/Web/MathML
| actusual wrote:
| Cool, why?
| valbaca wrote:
| "You should know" because, you should.
|
| (sarcastic b/c I had the same question)
| docfort wrote:
| I think it's better to know that sometimes we only know how to
| describe something by relating rates of change to other states.
| And that's ok. Maybe it has a closed form equation, or maybe can
| only be solved numerically. But if I see that a differential
| equation looks like a wave equation, then I get intuition that
| it's describing waves. And why do the waves appear? Because the
| physical process the PDE describes has a speed limit on
| information passing from time into space!
|
| Don't like traffic waves? Well, why is there some limit on
| spatial information connected to temporal information? It's
| because I cannot see through the cars in front of me. The "fog of
| war" creates the waves. The denser the fog (e.g. I'm surrounded
| by semitrucks), the greater the likelihood of waves developing.
|
| This intuition is formed by being able to recognize the form of
| the PDE with general knowledge of the solutions, without needing
| to actually solve the PDE. Sure, additional insights are possible
| if you solve it, but knowing that traffic is like springs gives
| you leverage to use your ordinary intuition to understand
| unfamiliar things.
|
| Point of fact, James Maxwell of E&M fame saw the wave equation
| and the separate electric and magnetic field PDEs and came up
| with a detailed spring model to give himself a more familiar
| analog to play with.
| dan-robertson wrote:
| To give Maxwell a little more credit (not that you aren't), the
| wave equations and PDEs of today are much nicer thanks to
| modern knowledge and computational techniques. Maxwell didn't
| have div, grad or curl and so he had dozens of equations to
| look at instead of just a few, and I think the terms and
| patterns weren't as well known as they are today.
| bernulli wrote:
| It's really cool how a Mach number emerges from traffic flow,
| with speed of cars vs speed of information, completely with
| shock waves and everything!
| [deleted]
| rq1 wrote:
| Black Scholes and Heat Equation are the same, up to a change of
| variable.
| groos wrote:
| One of them is not like the rest.
| mjfl wrote:
| what's the best black scholes tutorial?
| _se wrote:
| The "Natenberg Bible": https://www.amazon.com/Option-
| Volatility-Pricing-Strategies-...
| Extigy wrote:
| I too would have liked to have seen Navier-Stokes included, or
| least an inviscid Euler equation for modelling fluid flow.
| aaaaaaaaaaab wrote:
| To me the most baffling thing about differential equations is the
| fact that somehow the Universe is able to solve them in real
| time. I mean, of course there are PDEs like the Navier-Stokes
| equation that describe phenomena emerging from the simple
| interactions of an immense number of particles, so you could say
| that the Universe doesn't "solve" them per se, rather, it runs
| the discretized simulation on an extremely fine scale, and the
| whole continuous PDE is our "simplification" of the problem.
|
| However, there are equations like the Einstein field equations
| that operate on a seemingly continuous domain, and whose
| solutions are impossibly complex in nontrivial cases... So how
| does the Universe do it?
|
| One can say that this question is beyond what science should be
| concerned with; the Universe evolves according to these
| equations, because this is what the Universe _is_. Yet, from a
| computational point of view it irks me...
| [deleted]
| Koshkin wrote:
| > _the Universe is able to solve them in real time_
|
| Not just the Universe - analog computers can do that, too.
| Orangeair wrote:
| It's been awhile since I've had a Diff EQ class, but isn't the
| harmonic motion one an ODE?
| lucaspauker wrote:
| Yeah good catch
| ChrisRackauckas wrote:
| ODEs are one-dimensional PDEs.
| sampo wrote:
| The harmonic motion equation is an ordinary differential equation
| (ODE), not partial differential equation (PDE).
| The_rationalist wrote:
| fithisux wrote:
| Boltzmann equation?
| bally0241 wrote:
| Helmholtz equation?
| rudiger wrote:
| The Black-Scholes equation is basically identical to the heat
| equation. Divide through by s^2 and let n = s^2 * (T - t) if you
| want to derive it.
| dls2016 wrote:
| The Schrodinger equation is the heat equation with complex
| time. Although qualitatively it's dispersive, not dissipative.
| jcla1 wrote:
| The difference is that in the Schrodinger case you're
| effectively 'turning' the solution (in the complex plane)
| which leads to the uncomfortable question of whether the
| solution to the heat equation you'd start with is still
| defined. When going from heat to Black-Scholes you're just
| rescaling in 'existing' dimensions which doesn't change the
| character of the PDE.
| prof-dr-ir wrote:
| The author is an undergraduate student, and judging from this
| list it appears that he has yet to encounter non-linear PDEs?
|
| Besides the Navier-Stokes equations, which are already frequently
| mentioned, I would have very much liked to see Einstein's
| equations added as well.
| [deleted]
| brummm wrote:
| This seems like a random collection of equations from a 1st/2nd
| year undergrad physics class + Black Scholes.
| frakt0x90 wrote:
| I know I could look it up but having an explanation of what any
| of the variables mean would make this not useless.
| killjoywashere wrote:
| > an explanation of what any of the variables mean
|
| As I recall, that's pretty much Electricity & Magnetism 2 for
| physics undergrads. E.g.
| https://www.colorado.edu/sei/departments/physics/activities/...
| ccosm wrote:
| No Navier-Stokes? Elasticity?
| elil17 wrote:
| I don't really get the point of this.
|
| Who should know these?
|
| Why should they know them?
|
| What should they know about them?
|
| As a mechanical engineer, for instance, it's usually a bad idea
| for me to think about these equations - it's to "in the weeds",
| so to speak.
| JabavuAdams wrote:
| One of these is not like the others.
| pc86 wrote:
| Who is "you" in this scenario? Who actually needs to know these
| by heart day to day?
| valbaca wrote:
| Partial Differential Equations you should know but with no
| explanation...I mean...shouldn't you...just know? /s
| jvanderbot wrote:
| Pet peeve: Define your constants (at least units!). If I know the
| constants by heart, I probably remember the equation.
| dan-robertson wrote:
| Most of the units can be inferred, e.g. for the wave equation,
| say the units of _u_ are A (for amplitude, but you can guess
| whatever), _x_ is L (for length) and _t_ is T for time (both
| are extremely conventional dimensions). You convert the pde to
| units and get: A/T^2 = units(c)^2 A/L^2,
|
| and can therefore say: units(c) = L/T
|
| And guess that _c_ is the speed of the wave or something
| proportional to it (it is, in fact, the speed).
|
| For the simple harmonic oscillator you get:
| units(m) L/T^2 = units(k) L
|
| which is insufficiently determined but gives units(k/m) =
| 1/T^2, and you might guess m is mass (in kg say) and then k is
| kgs^-2, or force per distance, a reasonable set of units for a
| spring constant (the ode is just Hook's law: F=kl, but F=ma)
|
| For the other equations it becomes harder but the point of the
| website isn't really to teach you what the pde is. It's
| extremely easy to search for the equation on Wikipedia (as the
| site gives their names) and look up the units and a bit about
| the equations there.
___________________________________________________________________
(page generated 2022-02-08 23:00 UTC)